Problem: $ A = \left[\begin{array}{r}3 \\ 1\end{array}\right]$ $ B = \left[\begin{array}{rr}-1 & 0 \\ 4 & 2\end{array}\right]$ Is $ A- B$ defined?
Solution: In order for subtraction of two matrices to be defined, the matrices must have the same dimensions. If $ A$ is of dimension $( m \times  n)$ and $ B$ is of dimension $( p \times  q)$ , then for their difference to be defined: 1. $ m$ (number of rows in $ A$ ) must equal $ p$ (number of rows in $ B$ ) and 2. $ n$ (number of columns in $ A$ ) must equal $ q$ (number of columns in $ B$ Do $ A$ and $ B$ have the same number of rows? Yes Yes No Yes Do $ A$ and $ B$ have the same number of columns? No Yes No No Since $ A$ has different dimensions $(2\times1)$ from $ B$ $(2\times2)$, $ A- B$ is not defined.